Group #36441346 — subgroups

Subgroups — #36441346

Order 22784.

246 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151651	—	2	1	no	—
5	C₂×#15667948	C₄	1	2	yes	—
6	C₄×#13083558	—	2	1	no	—
7	#19151363	—	2	1	no	—
8	#19151351	C₄	1	2	yes	—
9	#19151349	—	2	1	no	—
10	C₂²×C₁₆×C₈₉	—	2	1	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₈₉	—	2	1	no	—
13	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
14	C₂×#13083558	—	2	2	no	—
15	#15667948	—	2	2	no	—
16	#15667948	C₈	1	4	yes	complemented
17	C₄⋊#10623494	—	2	2	no	—
18	C₂²⋊#10623494	—	4	1	no	—
19	#15667923	—	4	1	no	—
20	C₄×#10623494	—	4	1	no	—
21	C₄×#10623494	—	4	1	no	—
22	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	—	2	4	no	—
26	C₂×C₁₆×C₈₉	—	2	2	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
31	#13083558	—	2	4	no	—
32	#13083558	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
38	C₂×C₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
45	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₈₉	D₈	1	1	yes	characteristic
48	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
49	C₁₆×C₈₉	—	2	4	no	—
50	C₂³×C₈₉	#103	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#104	1	2	yes	—
59	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	C₄×C₈	1	1	yes	characteristic
64	C₈×C₈₉	#97	1	1	yes	characteristic
65	C₈×C₈₉	C₄⋊C₈	1	2	yes	—
66	#10623494	—	4	4	no	—
67	#10623494	—	8	2	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
71	C₂²×C₈₉	#340	1	1	yes	characteristic
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	D₄⋊C₈	1	2	yes	—
75	C₄×C₈₉	#336	1	1	yes	characteristic
76	C₄×C₈₉	#339	1	1	yes	characteristic
77	C₄×C₈₉	#359	1	2	yes	—
78	C₄×C₈₉	#368	1	2	yes	—
79	#7172	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	—	2	2	no	—
81	C₂×C₈₉	#1279	1	1	yes	characteristic
82	C₂×C₈₉	#1381	1	1	yes	characteristic
83	C₂×C₈₉	#1382	1	1	yes	characteristic
84	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
85	C₄×#108	—	89	1	no	—
86	#1564	—	89	1	no	—
87	C₈₉	#7172	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	2	1	no	—
89	#431	—	178	1	no	—
90	C₂×#108	—	89	2	no	—
91	C₄×M₁₆	—	178	1	no	—
92	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₄×C₁₆	—	2	1	no	—
94	C₄×C₁₆	Dic₈₉	1	2	yes	—
95	#341	—	178	1	no	—
96	#335	—	89	2	no	—
97	#334	—	178	1	no	—
98	C₂×M₁₆	—	178	2	no	—
99	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
101	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	#108	—	89	4	no	—
105	#108	—	178	2	no	—
106	C₄⋊C₈	—	178	2	no	—
107	C₂²⋊C₈	—	356	1	no	—
108	#97	—	356	1	no	—
109	C₄×C₈	—	356	1	no	—
110	C₄×C₈	—	356	1	no	—
111	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
112	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
115	C₂²×C₄	#13083551	1	1	yes	characteristic
116	C₂²×C₄	#13083563	1	1	yes	characteristic
117	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
118	M₁₆	—	356	2	no	—
119	M₁₆	—	178	4	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	356	2	no	—
122	C₂×C₈	—	356	2	no	—
123	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
125	C₂×C₈	#13083553	1	1	yes	characteristic
126	C₂×C₈	#13083553	1	1	yes	characteristic
127	C₂×C₈	C₈×D₈₉	1	2	yes	—
128	C₂×C₈	#13083557	1	2	yes	—
129	C₄²	#13083551	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
131	C₄²	D₇₁₂	1	1	yes	characteristic
132	C₄²	#13083563	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂³	#15667937	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#15667919	1	1	yes	characteristic
138	C₂×C₄	#15667924	1	2	yes	—
139	C₂×C₄	#15667934	1	1	yes	characteristic
140	C₂×C₄	#15667934	1	1	yes	characteristic
141	C₂×C₄	#15667937	1	1	yes	characteristic
142	C₂×C₄	#15667940	1	2	yes	—
143	C₂×C₄	#15667944	1	1	yes	characteristic
144	C₂×C₄	#15667946	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	356	4	no	—
148	C₈	—	712	2	no	—
149	C₈	C₈×Dic₈₉	1	1	yes	characteristic
150	C₈	#15667922	1	1	yes	characteristic
151	C₈	#15667941	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	#19151330	1	2	yes	—
155	C₂²	#19151361	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#19151327	1	2	yes	—
159	C₄	#19151352	1	1	yes	characteristic
160	C₄	#19151358	1	1	yes	characteristic
161	C₄	#19151406	1	2	yes	—
162	C₄	#19151428	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#26018092	1	1	yes	characteristic
165	C₂	#26018346	1	1	yes	characteristic
166	C₂	#26018348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15667948

C₂

yes

maximal, characteristic

#26019000

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

#19151651

—

C₂×#15667948

C₄

yes

—

C₄×#13083558

—

#19151363

—

#19151351

C₄

yes

—

#19151349

—

C₂²×C₁₆×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₄

yes

—

C₂×#13083558

—

#15667948

—

#15667948

C₈

yes

—

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

Q₈

yes

—

—

—

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

SD₁₆

yes

characteristic

C₂²×C₄×C₈₉

Q₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

#103

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×C₈

yes

characteristic

C₈×C₈₉

#97

yes

characteristic

C₈×C₈₉

C₄⋊C₈

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

Q₈⋊C₈

yes

—

C₂²×C₈₉

#340

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

D₄⋊C₈

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₄×C₈₉

#368

yes

—

#7172

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₄×C₁₆

D₈₉

yes

characteristic

C₄×#108

—

#1564

—

C₈₉

#7172

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

178

—

C₂×#108

—

C₄×M₁₆

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

Dic₈₉

yes

—

#341

—

178

—

#335

—

#334

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

100

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

101

C₂×C₄²

D₃₅₆

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

—

178

—

106

C₄⋊C₈

—

178

—

107

C₂²⋊C₈

—

356

—

108

#97

—

356

—

109

C₄×C₈

—

356

—

110

C₄×C₈

—

356

—

111

C₄×C₈

Dic₁₇₈

yes

characteristic

112

C₄×C₈

C₄×D₈₉

yes

characteristic

113

C₄×C₈

C₂×Dic₈₉

yes

characteristic

114

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

115

C₂²×C₄

#13083551

yes

characteristic

116

C₂²×C₄

#13083563

yes

characteristic

117

C₂²×C₄

Dic₃₅₆

yes

characteristic

118

M₁₆

—

356

—

119

M₁₆

—

178

—

120

C₂×C₈

—

121

C₂×C₈

—

356

—

122

C₂×C₈

—

356

—

123

C₂×C₈

C₄×Dic₈₉

yes

characteristic

124

C₂×C₈

C₂²⋊Dic₈₉

yes

characteristic

125

C₂×C₈

#13083553

yes

characteristic

126

C₂×C₈

#13083553

yes

characteristic

127

C₂×C₈

C₈×D₈₉

yes

—

128

C₂×C₈

#13083557

yes

—

129

C₄²

#13083551

yes

characteristic

130

C₄²

C₄⋊Dic₈₉

yes

characteristic

131

C₄²

D₇₁₂

yes

characteristic

132

C₄²

#13083563

yes

characteristic

133

C₁₆

—

134

C₂³

#15667937

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#15667919

yes

characteristic

138

C₂×C₄

#15667924

yes

—

139

C₂×C₄

#15667934

yes

characteristic

140

C₂×C₄

#15667934

yes

characteristic

141

C₂×C₄

#15667937

yes

characteristic

142

C₂×C₄

#15667940

yes

—

143

C₂×C₄

#15667944

yes

characteristic

144

C₂×C₄

#15667946

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

356

—

148

C₈

—

712

—

149

C₈

C₈×Dic₈₉

yes

characteristic

150

C₈

#15667922

yes

characteristic

151

C₈

#15667941

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

#19151330

yes

—

155

C₂²

#19151361

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

#19151327

yes

—

159

C₄

#19151352

yes

characteristic

160

C₄

#19151358

yes

characteristic

161

C₄

#19151406

yes

—

162

C₄

#19151428

yes

—

163

C₂

—

164

C₂

#26018092

yes

characteristic

165

C₂

#26018346

yes

characteristic

166

C₂

#26018348

yes

characteristic

	subgroup	G/H
center	C₄²	D₇₁₂
commutator	C₈×C₈₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#103
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#7172
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₄²

D₇₁₂

commutator

C₈×C₈₉

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₈₉

Fitting

C₂×C₄×C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄×C₈×C₈₉

C₂²

hypercenter

C₂×C₄×C₁₆

D₈₉

hypocenter

C₈₉

#7172

LCS

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
16. #15667948	C₈	semidirect
87. C₈₉	#7172	semidirect

H ≅ G/N

type

16. #15667948

C₈

semidirect

87. C₈₉

#7172

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#15667948	2
#26019000	2
C₂×C₄×C₁₆×C₈₉	2
#7172	89

subgroup

index

C₄×#15667948

#26019000

C₂×C₄×C₁₆×C₈₉

#7172

Subgroups — #36441346

Conjugacy classes of subgroups

Complements

Maximal subgroups